Artificial boundary conditions for Stokes and Navier-Stokes systems in layer-like domains (Q2455312)

The authors deal with the following stationary problem \[ -\nu\Delta_x u+(u\cdot\nabla) u+\nabla_x p= f\quad\text{in }\Omega, \] \[ \nabla_x u= 0\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a domain with smooth boundary \(\partial\Omega\) coinciding with the layer \(\mathbb{B}_R= \{x\in\mathbb{R}^3\mid|x|< R\}\) of unit thickness outside the ball \(B_*= \{x= (y,z)\mid y= (y_1, y_2)\in\mathbb{R}^2\), \(|z|<{1\over 2}\}\). Here \(u= (u_1, u_2, u_3)\) and \(p\) are the velocity and pressure, respectively; \(\nu> 0\) is the constant viscosity of the fluid. The authors specify mixed-type artificial boundary conditions on the cylindrical cutting surface \(\Gamma_R\), \(T_R= \{x\mid|y|= R\), \(|z|< 1\}\) that ensure a superpower accuracy of approximation for Stokes problem and the error \(O(R^{-3})\) for Navier-Stokes problem.